CAREER: Geometric Potential Theory

该职业奖将支持多方面的研究和教育计划，旨在这两个领域取得重大进展。研究目标围绕着对不均匀曲率形状的更深入理解。平面上两点之间的距离由连接它们的线段的长度给出。线段可以用尺子来构造，但是在任意几何形状上找到两点之间的距离要困难得多，部分原因是这种形状没有尺子！找到这个“标尺”，即，测量几何形状上距离的最佳方法，植根于数学物理的深层问题。在数学中，我们使用度量来测量距离。当试图找到理想的度量时，人们通常必须找到一个光滑的函数来解决特定的偏微分方程。这是一个具有作用泛函的优化问题，其极小值恰好是偏微分方程的解。可以将非光滑函数插入动作泛函中，称为势，打开了通常被称为潜在方程的势理论的大门。这个项目涉及复杂几何中的问题，其中考虑的潜力可以被赋予一个非常具体的度量几何，导致比平常更微妙的理解。除了上述研究之外，本项目还将实施垂直整合的教育计划，包括各种形式的STEM领域的公共推广（如制作教育视频并将其发布到网上）、本科生暑期研究以及研究生的参与。本项目的研究目标可以分为三个部分。 第一部分讨论Kahler度量空间中测地线的几何势理论，并对复几何中正则Kahler度量的存在性进行了各种刻画。随着测地线射线的度量几何的充分发展，人们可以将这些几何视为测地线射线空间上的优化问题，并考虑对射线的正则性进行各种改进。第二部分致力于奇异型空间的几何势理论，着眼于具有规定的奇异性和乘子理想层的变化的复杂Monge-Ampere方程。奇异类型的度量空间预计是完整的，它将允许奇异的Kahler-Einstein度量的研究，在奇异性的变化。在最后一部分中，我们研究了Kahler几何与几何分析的其他部分（包括Hermitian几何和凸分析）的相互作用。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This CAREER award will support a multifaceted program of research and education aiming at significant progress in both areas. The research goals center around a deeper understanding of shapes with uneven curvature. The distance between two points in the plane is given by the length of the segment joining them. The segment can be constructed with a ruler, however finding the distance between two points on an arbitrary geometric shape is much more difficult, partly because such shapes don't come with a ruler! Finding this "ruler", i.e., the best possible way to measure distances on geometric shapes, is rooted in deep problems of mathematical physics. In mathematics we measure distances using metrics. When trying to find ideal metrics, one often has to find a smooth function that solves a specific partial differential equation. This is an optimization problem with an action functional whose minimizers are exactly the solutions of the partial differential equation. It is possible to plug in non-smooth functions into the action functional, called potentials, opening the door to what is often referred to as the potential theory of the underlying equation. This project deals with problems in complex geometry where the potentials considered can be given a very specific metric geometry, leading to a much more delicate understanding than usual. In addition to the proposed research, the project will pursue a vertically integrated educational program that includes various forms of public outreach popularizing STEM fields (such as creating educational videos and posting them online), conducting undergraduate summer research and the involvement of graduate students.The research goals of the project can be split in three parts. The first part is devoted to the geometric potential theory of the geodesic rays inside the space of Kahler metrics, with a view toward various characterizations for existence of canonical Kahler metrics in complex geometry. With the metric geometry of geodesic rays sufficiently developed, one can look at these conjectures as optimization problems on the space of geodesic rays, with various refinements on the regularity of the rays considered. The second part is devoted to the geometric potential theory of the space of singularity types, with a view toward complex Monge-Ampere equations with prescribed singularity and variation of multiplier ideal sheaves. The metric space of singularity types is expected to be complete, and it will allow for a study of singular Kahler-Einstein metrics, under variation of the singularity. In the last part we study interactions of the investigations in Kahler geometry with other parts of geometric analysis, including Hermitian geometry and convex analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.